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Parallel Implementation of an Integral Wavefield ExtrapolatorYanpeng Mi* and Gary F. Margrave, CREWES Project, Geology and Geophysics Department,

The University of Calgary, 2500 University Drive, Calgary, Alberta T2N 1N4, Canada

Summary

Highly accurate seismic depth imaging (migration) is required by the oil and gas exploration industry in regions of complex geology. Fourier-

domain depth-imaging techniques provide the needed accuracy although the computation cost can be large when the lateral velocity

variation is rapid. Parallel computer network provide an efficient solution to overcome the computational barrier of Fourier-domain imaging

techniques. In this paper, a new Fourier-domain imaging technique, based on nonstationary filtering and wavefield extrapolation theories, is

described. The parallel version of the algorithm is implemented using C++ and Fortran 90 on an Alpha cluster workstations. The Message

Passing Interface (MPI) library was used for data distribution and collection. The algorithm, found to be accurate but slow on serial

machines, can achieve a speed acceptable for most industrial applications.

Introduction

Seismic depth imaging repositions the scattered energy from the subsurface to its correct spatial location according to a velocity model.

Produced by a true-amplitude depth imaging technique, a subsurface image also carries amplitude information that is directly proportional

to reflection coefficients.

Kirchhoff depth-imaging techniques are very common in the exploration industry due to robustness and high efficiency. Theoretically the

method is based on Green�s theorem and it is implemented by summation along iso-time surfaces computed by raytracing. Difficulties arise

when it is used in regions of complex geology. Fourier-domain imaging algorithms, especially for complex media, are often slower than

Kirchhoff-type techniques. However, they provide a higher accuracy solution by allowing the energy to propagate in all possible directions

instead of only the Snell�s law paths. Fourier-domain techniques can be made efficient by making reasonable approximations and operating

in efficient data domains.

Fourier domain wavefield extrapolation and integral wavefield extrapolators

The wavefield at depth z can be obtained by extrapolation the wavefield at depth 0, given knowledge of the velocity field between 0 and z.

Consider a mono-frequency wavefield at depth 0, � ��� ,0,k x �z , which has been 2D forward Fourier transformed to frequency-

wavenumber (�,kx) domain, the wavefield at depth z can be written as (Gazdag, 1978),

� � � � ),,,k(,0,k,,k xxx ������ zz � ,(1)

where 2

2

2

),,k( x

xkv

iz

ez��

�

�

�� is the (�,kx) domain wavefield extrapolator operating from 0 to z. The plus and minus signs denote

upward and downward extrapolation when the z-axis points downward. For heterogeneous media, the extrapolation operator is known only

approximately and the step size is limited. The extrapolation is done recursively with small steps, so that local wave propagation can be

approximated by homogeneous and isotropic propagation theory.

There are several ways to compute an approximate wavefield extrapolator for heterogeneous media. Typical algorithms are phase-shift-

plus-interpolation (PSPI) originally developed by Gazdag and Sguazzero (1984) and the split-step Fourier algorithm originally developed by

Stoffa et al. (1990), which is also called the phase-screen algorithm by Wu and Huang (1992). There are many further developments of the

above algorithms in order to deal with extreme lateral velocity variation and achieve high efficiency (Jin and Wu, 1998; Popovici, 1996; etc.)

These algorithms have also been implemented on parallel computers, for example, Tanis and Stoffa (1997).

Nonstationary Fourier wavefield extrapolators

Using a complete set of reference velocities, the PSPI algorithm becomes an integral over horizontal wave number kx, which performs

wavefield extrapolation simultaneously with an inverse Fourier transform (Margrave and Ferguson 1999a)

� �� �

� � � � � ����

��

�� xxxx2kkexp,0k,kx

2

1,,x dxi,,z,z ����

�

�� ,

(2)

where � �2

)(2

2

x ,kxxk

xviz

e,z,��

�

�

�� is the nonstationary wavefield extrapolator. � ��� ,,x z is the (�,x) domain expression of the

wavefield at depth z and � ��� ,0kx , is the (�,kx) domain expression of the wavefield at depth 0. Though there is no interpolation in

equation (2), the expression PSPI is used for equation (2) because it is a limiting form of Gazdag�s PSPI.

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Equation (2) has a complementary form, called nonstationary phase-shift (NSPS), which performs wavefield extrapolation simultaneously

with a forward Fourier transform (Margrave and Ferguson, 1999a)

� � � � � � � ����

��

� xkexp,0x,kx,,k xxx dxi,,z,z ������ ,(3)

where � ��� ,kx x ,z, is expressed as before.

Equation (2) can be approximately computed by matrix-vector multiplication, which can be written as

0�� A

z� ,

(4)

where �0 and �

z are column vectors representing a mono-frequency wavefield in (�,k

x) domain at depth 0 and the extrapolated wavefield

in (�,x) domain at depth z, respectively. Matrix A is the combination of the wavefield extrapolator and the simultaneous inverse Fourier-

transform kernel

��������

��������

�

������

��

���

����

�

���

���

��

���

����

�

���

���

xnnxnn

xnxn

xnxnxx

kxkv

zikxkv

zi

kxkv

zikxkv

zi

ee

ee

A

22

21

212

2

12

21

211

212

1

2

...

.....

.....

...

��

��

. (5)

Similarly, integral (3) can also be approximated by matrix-vector multiplication, however with kx varying in the column direction while x

varying in the row direction.

Margrave and Ferguson (1999b) showed that the NSPS and PSPI can be naturally combined into a symmetric wavefield extrapolator

(SNPS) by first performing NSPS for the upper half z and PSPI for the lower half z within a single step. A Taylor series derivation of PSPI

and NSPS and related error analysis showed that the first-order errors of PSPI and NSPS oppose one another, so that SNPS has a smaller

error and is more stable than either PSPI or NSPS alone (Margrave and Ferguson, 2000).

Parallel implementation on MACI Alpha Cluster

The integral SNPS extrapolation algorithm was implemented on the Multimedia Advanced Computational Infrastructure (MACI) Alpha

Cluster at the University of Calgary. The cluster consists of 128 Compaq Alpha workstations and each single user can use up to 16 CPUs

for a single computing task. A general network configuration of the Alpha Cluster is shown in figure 1.

The Message Passing Interface (MPICH1.2) was used for parallel implementation. Each computing node was assigned a single shot

gather migration task. It took about 8 hours to migrate the 240 shot gathers on 16 XP1000 workstations. Figure 2 shows the band-

limited (0-20 Hz) reflectivity and the subsurface image computed by the integral SNPS.

Figure 1. The general network configuration of the Alpha Cluster at the University of Calgary.

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Conclusions

Integral SNPS extrapolator produces very accurate subsurface image. Direct computation of equation (2) and (3) can be made faster by

utilizing the symmetric properties of the extrapolator and the forward/inverse Fourier transform kernels. Faster while less accurate

numerical algorithms to compute the square root and exponential function can also be used to make the algorithm more efficient.

Acknowledgements

We would like to thank the colleagues of the MACI project for their support.

(a) (b)

Fig. 2. (a) 0-20 Hz band-limited reflectivity of the Marmousi model and (b) the subsurface image computed by the integral SNPS

REFERENCES

Ferguson, R. and Margrave, G., 1999, A practical implementation of depth migration by nonstationary phase shift: Annual Meeting

Abstracts, Society Of Exploration Geophysicists, 1370-1373.

Gazdag, J. and Sguazzero, P., 1984, Migration of seismic data by phase shift plus interpolation, Geophysics, 49, 124-131.

Gazdag, J., 1978, Wave equation migration with the phase-shift method, Geophysics, 43,1342-1351.

Jin, S. and Wu, R., 1998, Depth migration using the windowed generalized screen propagators: Annual Meeting Abstracts, Society Of

Exploration Geophysicists, 1843-1846.

Margrave, G. F. and Ferguson, R. J., 1999a, Wavefield extrapolation by nonstationary phase shift: Geophysics, 64, no. 04, 1067-1078.

Margrave, G. and Ferguson, R., 1999b, An explicit, symmetric wavefield extrapolator for depth migration: Annual Meeting Abstracts,

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Margrave, G. and Ferguson, R., 2000, Taylor series derivation of nonstationary wavefield extrapolators: Annual Meeting Abstracts, Society

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Popovici, A. M., 1996, Prestack migration by split-step DSR: Geophysics, 61, no. 05, 1412-1416.

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